I have a cross-sectional data set with about 8000 observations on child obesity (eg BMI). This data was collected in 8 countries and within schools (about 200 schools), i.e. observations are clustered within schools and countries. I am interested in how child characteristics (e.g. socio economic status) relate to child obesity. My approach was to estimate a pooled OLS and clustering at the school level and also including country dummy variables in the regressions. As I have many and reasonably large school clusters, I thought this would be a good approach. I have however been told that I should estimate a FE effect model. Is it possible to estimate a country fixed-effects model and cluster at the school level?
I think it is doable; this would be one of the relatively rare examples where the assumptions on the sample sizes will be met sufficiently well. Most education researchers would want to fit a classic multilevel model with three levels, and then I would say that 8 countries are not enough to reliably estimate variances at level 3. Depending on the software, you might be able to mix fixed effects with clustered standard errors (Stata
xtreg ..., fe vce(cluster ...)), or you may have to include the country dummies as regressors to implement the country fixed effects.
You may also want to try some country-school or country-individual interactions, as TV programs in North Korea may not be so interesting to watch as in the US, so kids would spend less time in front of TV in some countries.